<s>
A	O
simultaneous	B-Algorithm
eating	I-Algorithm
algorithm	I-Algorithm
(	O
SE	O
)	O
is	O
an	O
algorithm	O
for	O
allocating	O
divisible	O
objects	O
among	O
agents	O
with	O
ordinal	O
preferences	O
.	O
</s>
<s>
To	O
do	O
this	O
,	O
the	O
n-by-n	O
matrix	O
of	O
probabilities	O
should	O
be	O
decomposed	O
into	O
a	O
convex	O
combination	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
done	O
by	O
the	O
Birkhoff	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
It	O
is	O
guaranteed	O
to	O
find	O
a	O
combination	O
in	O
which	O
the	O
number	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
is	O
at	O
most	O
n2-2n	O
+2	O
.	O
</s>
<s>
Note	O
that	O
the	O
random	B-Algorithm
priority	I-Algorithm
rule	O
,	O
which	O
solves	O
the	O
same	O
problem	O
as	O
PS	O
,	O
is	O
truthful	O
.	O
</s>
<s>
The	O
algorithm	O
is	O
based	O
on	O
repeatedly	O
solving	O
instances	O
of	O
parametric	O
network	B-Algorithm
flow	I-Algorithm
.	O
</s>
<s>
The	O
algorithm	O
uses	O
as	O
subroutines	O
both	O
the	O
PS	O
algorithm	O
and	O
the	O
Birkhoff	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
A	O
polynomial-time	O
algorithm	O
for	O
computing	O
allocations	O
that	O
are	O
ex-ante	O
proportional	O
,	O
and	O
ex-post	O
both	O
PROP1	O
and	O
1/2	O
-fraction	O
maximin-share	B-Algorithm
(	O
and	O
also	O
1/2	O
-fraction	O
truncated-proportional	O
share	O
)	O
.	O
</s>
